On the coarse geometry of solvable Baumslag-Solitar groups and relatively hyperbolic groups

Thesis

Firstly, we prove that the solvable Baumslag--Solitar groups are rigid under quasiisometric embeddings, strengthening a classic result of Farb and Mosher. These same ideas allow us to give a quasiisometric classification of treebolic spaces, confirming a conjecture of Woess. Both of these results are proved by relating the boundedness of a novel integer sequence X(p,q,p',q') to a pair of treebolic spaces.

In the second chapter, which is joint with Sam Hughes and Davide Spriano, we prove that the language of (lambda',0)-quasigeodesics in a non-hyperbolic group is not regular for lambda' > 54. This is a strong converse to a result of Holt and Rees in which they prove that in a hyperbolic group the (lambda,epsilon)-quasigeodesics are regular whenever lambda is rational. So we have provided a new characterisation of hyperbolic groups in terms of whether their quasigeodesics form regular languages.

In the next chapter, inspired by Buyalo, Dranishnikov and Schroeder's Alice's Diary, we develop a general theory of diaries and linear statistics. These notions provide a powerful framework by which one can take a quasiisometric embedding of a metric space into a product of infinite-valence trees and upgrade it to a quasiisometric embedding into a product of binary trees.

Consequently, in the final chapter, we use diaries and linear statistics to prove that if a group G is relatively hyperbolic with respect to virtually abelian peripheral subgroups then G quasiisometrically embeds into a product of binary trees. This extends the result of Buyalo, Dranishnikov and Schroeder in which they prove that a hyperbolic group quasiisometrically embeds into a product of binary trees. To prove this result, we rely on the machinery of projection complexes and quasi-trees of metric spaces developed by Bestvina, Bromberg, Fujiwara and Sisto. We build on this theory by proving that one can remove certain edges from the quasi-tree of metric spaces, and be left with a tree of metric spaces which is quasiisometric to the quasi-tree of metric spaces. In particular, this reproves a result of Hume.

Authors: Nairne, P

• DOI: 10.5287/ora-2axv1qvdr
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: Baumslag-Solitar groups; Quasiisometric rigidity; geometric group theory; Quasiisometric embeddings; regular languages
• Subjects: Mathematics